With the economic outlook weakening, they argue that big deficits are looming for the next academic year and that they need to preserve the funds to prevent future layoffs. Los Angeles, for example, is projecting a $280 million budget shortfall next year that could threaten more jobs.

“You’ve got this herculean task to deal with next year’s deficit,” said Lydia L. Ramos, a spokeswoman for the Los Angeles Unified School District, the nation’s second-largest after New York City.

“So if there’s a way that you can lessen the blow for next year,” she said, “we feel like it would be responsible to try to do that.”

The district laid off 682 teachers and counselors and about 2,000 support workers this spring and was not sure it would be able to hire any of them back with the stimulus money. The district says it could be forced to cut 4,500 more people next year.